Chapter 2 Notes.notebook
October 17, 2012
2.1 Conditional Statements
Conditional: An if­then statement.
If you are not completely satisfied, then your money will be refunded.
Hypothesis
Conclusion
Every conditional has two parts: Hypothesis and Conclusion
Hypothesis: The part following if
Conclusion: The part following then
If you want to be fit, then you want to get plenty of exercise.
If T ­ 15 = 23, then T = 38.
Hypothesis
Conclusion
Writing a conditional
A rectangle has four right angles.
If a figure is a rectangle
then it has four right angles
An integer that ends with 0 is divisible by 5.
If
an integer ends with 0
then it is divisible by 5.
Chapter 2 Notes.notebook
October 17, 2012
A conditional can have a truth value of true or false.
To show that a conditional is true:
You have to show that every time the hypothesis is true, the conclusion is also true.
To show that a conditional is false:
You need to find only one counterexample for which the hypothesis is true and the conclusion is false.
Show that the following conditional is false by finding a counterexample.
If it is February, then there are 28 days in the month.
Leap year
Odd integers less than 10 are prime.
9
Venn Diagrams
If you live in Missoula, then you live in Montana.
Montana
Missoula
Carrots are vegetables.
Chapter 2 Notes.notebook
October 17, 2012
Converse
The converse of a conditional switches the hypothesis and the conclusion.
Conditional
If two lines intersect to form right angles, then they are perpendicular.
Converse
If two lines are perpendicular, then
they intersect to form right angles.
Find the converse of the following conditional
If two lines do not intersect and are not parallel, then they are skew.
If two lines are skew, then
they do not intersect and are not parallel
It is possible for a conditional and a converse to have the same or different truth values.
Conditional
If a figure is a square, then it has four sides.
Converse
If a figure has 4 sides,
then it is a square.
ConditionalIf p, then q
ConverseIf q, then p
Truth Value
Chapter 2 Notes.notebook
October 17, 2012
2.2 Biconditionals and Definitions
When a conditional and its converse are true, you can combine them as a true biconditional.
Join the two with if and only if
Conditional
If two angles have the same measure, then the angles are congruent.
Converse
If two angles are congruent, then the angles have the same measure.
Biconditional
Two angles have the same measure if and only if the angles are congruent.
If 3 points lie on the same line, then they are collinear.
If 3 points are collinear, then
they lie on the same line.
3 points are collinear if and only if they lie on
the same line.
If x = 7, then IxI = 7
Chapter 2 Notes.notebook
October 17, 2012
Definitions
A good definition has several components:
Uses clearly understood terms (commonly understood or already defined)
Precise (avoid words such as large, sort of and almost)
Reversible (can be written as a biconditional)
Chapter 2 Notes.notebook
October 17, 2012
Chapter 2 Notes.notebook
October 17, 2012
2.3 Deductive Reasoning
Deductive Reasoning is Logical Reasoning
A car mechanic knows:
If a car has a dead battery, then the car won't start.
The mechanic notices that a car has a dead battery.
The mechanic can use deductive reasoning (logic) to determine that the car will not start.
Law of Detachment
If a conditional is true and its hypothesis is true, then its conclusion is true.
If is a true statement and p is true, then q is true.
What can you conclude?
Given:
• If M is the midpoint of a segment, then it divides the segment into two congruent segments.
• M is the midpoint of Chapter 2 Notes.notebook
October 17, 2012
Law of Syllogism
If and are true statements, then
is a true statement
What can you conclude?
Given:
If a number ends in 0, then it is divisible by 10.
If a number is divisible by 10, then it is divisible by 5.
Chapter 2 Notes.notebook
October 17, 2012
2.4 Reasoning in Algebra
Justifying Steps in Solving an Equation
(needed when doing proofs)
Find x. Justify each step.
Solve for x. Justify each step.
Reason
AC=21
Chapter 2 Notes.notebook
October 17, 2012
Chapter 2 Notes.notebook
October 17, 2012
Chapter 2 Notes.notebook
October 17, 2012
2.5 Proving Angles Congruent
Theorem: A conjecture that is proven true.
Use postulates, definitions, and properties to prove theorems.
We will use Two­column proofs
When proving a theorem always start with the "Given" information.
Given information can be a statement or any information you can see on a diagram.
Given:
and
are vertical angles.
1
Prove: Statement
and
3
are vertical angles.
Reason
Given
2
Chapter 2 Notes.notebook
Statement
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Reason
Given
Chapter 2 Notes.notebook
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a, d
b
a
b, c
c
l
d
j, k
f
i, f
g
h
g, i
i
g, f
k
j
i, e
l, m
Find x.
e
m
Chapter 2 Notes.notebook
October 17, 2012
Chapter 2 Notes.notebook
October 17, 2012
Prove: 2 3
Statement
Reason
Given: AVC DVB
Prove: AVB DVC Statement
Reason
Chapter 2 Notes.notebook
Statement
Statement
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Reason
Reason
Chapter 2 Notes.notebook
October 17, 2012